Deeper than primes

[doronshadmi]

Length is a function of 2 points and location is a function of 1 point. So 2-1 = 1-0 ==> aleph1, aleph 0.

All points must be located in k-space, otherwise no action.

Length 0 is Length exactly as cardinality 0 is Cardinality.

So, your "Length is a function of 2 points" has no impact on the difference between Cardinality and Length, if the the considered values are not 0.

Again.

Aleph1 is Cardinality, which is the number of the objects that is the same for the different line segments, where Length (between the considered line segments) is different, as shown in the following diagram:

So once again, please use Traditional Math in order to rigorously define the difference between Cardinality (which is the same for the different line segments) and Length (which is not the same for the considered line segments).

 

[epix]

So once again, please use Traditional Math in order to rigorously define the difference between Cardinality (which is the same for the different line segments) and Length (which is not the same for the considered line segments).

A) In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

B) In geometric measurements, length most commonly refers to the longest dimension of an object.

Therefore Difference = A - B

 

[doronshadmi]

A) In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

B) In geometric measurements, length most commonly refers to the longest dimension of an object.

Therefore Difference = A - B

Traditional Math claims that line segments with different length are completely covered by sets of distinct points with the same Cardinality.

Please provide the formal proof of that claim.

 

[epix]

Traditional Math claims that line segments with different length are completely covered by sets of distinct points with the same Cardinality.

Please provide the formal proof of that claim.

No, that's what you claim. The definition says that there is no location on the line segment where a point couldn't be drawn. Hence "completely covered."

 

[doronshadmi]

No, that's what you claim. The definition says that there is no location on the line segment where a point couldn't be drawn. Hence "completely covered."

If there is no location on the line segment where a point couldn't be located, then there are no gaps between the distinct points of a given set of points, which are located along a given line segment, no matter what length > 0 that line segment has.

Please provide the formal mathematical proof of line segments with different lengths, which have such sets of points along them with the same cardinality.

 

[epix]

If there is no location on the line segment where a point couldn't be located, then there are no gaps between the distinct points of a given set of points, which are located along a given line segment, no matter what length > 0 that line segment has.

Yes, that's called the continuum.

Please provide the formal mathematical proof of line segments with different lengths, which have such sets of points along them with the same cardinality.

Your request doesn't make any sense, coz it's based on your phantasmagorical definition of point/member of R. You think that if there is no gap between points on the real line, the points are necessarily adjacent, but the concept of adjacency doesn't exist with 0-dim objects. That's one reason why there are "no gaps," coz non-adjacent means a space between two objects: If the concept of adjacency cannot be applied to the spacial relationship, so cannot the concept of "gaps."

Your embedded experience acquired by peeling potatoes in the kitchen is an insurmountable obstacle to navigate through the topic.

 

[doronshadmi]

Yes, that's called the continuum.

Your request doesn't make any sense, coz it's based on your phantasmagorical definition of point/member of R. You think that if there is no gap between points on the real line, the points are necessarily adjacent, but the concept of adjacency doesn't exist with 0-dim objects. That's one reason why there are "no gaps," coz non-adjacent means a space between two objects: If the concept of adjacency cannot be applied to the spacial relationship, so cannot the concept of "gaps."

Your embedded experience acquired by peeling potatoes in the kitchen is an insurmountable obstacle to navigate through the topic.

Your reply makes no sense as follows:

1) If the concept of adjacency doesn't exist with 0-dim objects, then there exists one and only one 0-dim element, because all locations have 0 distance between them (which means one and only one location).

2) Let us say that by some miracle there are many distinct 0-dim elements that have no gaps between them. In that case you still have to provide the formal mathematical proof of line segments with different lengths, which have sets of points with no gaps between them, where those sets have the same cardinality.

 

[epix]

Your reply makes no sense as follows:

1) If the concept of adjacency doesn't exist with 0-dim objects, then there exists one and only one 0-dim element, because all locations have 0 distance between them (which means one and only one location).

2) Let us say that by some miracle there are many distinct 0-dim elements that have no gaps between them. In that case you still have to provide the formal mathematical proof of line segments with different lengths, which have sets of points with no gaps between them, where those sets have the same cardinality.

You continue to apply your kitchen experience. You should check in for the summer season: http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

 

[doronshadmi]

You think that if there is no gap between points on the real line, the points are necessarily adjacent,

Wrong. If there are no gaps between points, then there is one and only one point.

Actually by your reasoning multiplicity is impossible, no matter what dim is used.

You continue to apply your kitchen experience. You should check in for the summer season: http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Please provide the formal mathematical proof for the existence of two 0-dim objects, such that "adjacency doesn't exist with 0-dim objects" is an essential part of that proof.

The stage is yours.

 

[doronshadmi]

In general it has to be stressed that existing things (where dim elements are some particular case of existing things) do not have multiple elements (finite or infinite) if only the same dim is used.

For example:

If only 0-dim level is used, then there is only one 0-dim element.

If only 1-dim level is used, then there is only one 1-dim element.

...

If only n-dim level is used, then there is only one n-dim element.

---------------

In other words, multiple elements are defined only by co-existence of more than single dim level.

By the co-existence the greater dim level is non-local w.r.t to the smaller dim level , exactly because it is irreducible to the smaller dim level (otherwise smaller and greater dims do not exist, in the first place).

By the co-existence the smaller dim level is local w.r.t to the greater dim level, exactly because it is non-extendible to the greater dim level (otherwise smaller and greater dims do not exist, in the first place).

 

[epix]

Wrong. If there are no gaps between points, then there is one and only one point.

According to your kitchen experience.

Please provide the formal mathematical proof for the existence of two 0-dim objects, such that "adjacency doesn't exist with 0-dim objects" is an essential part of that proof.

The stage is yours.

 

[doronshadmi]

According to your kitchen experience.



[qimg]http://img685.imageshack.us/img685/8656/proofqh.jpg[/qimg]

Homeomorphism is a bidirectional function which enables deformation of a given space without making separations of that space.

Since separation is essential for the existence of more than one 0-dim element, there can't be a bidirectional function which (enables deformation between 0-dim and 1-dim) AND (also enables the existence of no more then one 0-dim element).

Furthermore, stretched 0-dim is not 0-dim anymore, because we get at least 1-dim.

Also a totally reducible 1-dim is not 1-dim anymore, because we get at most 0-dim.

In other words, your quote is a load of nonsense (and once again you do not provide the origin of this quote, as if it is your original work ( it was taken from http://www.math.wisc.edu/~miller/res/map.pdf , which is Arnold W. Miller's work )).


--------------

Let us use WikiPedia ( http://en.wikipedia.org/wiki/Homeomorphism ):

A continuous deformation between a coffee mug and a doughnut illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic—only a continuous mapping with a continuous inverse.

Please formally define a continuous mapping with a continuous inverse between, at least, 0-dim and 1-dim (please be aware that the example above is done at the same dim, which is 3-dim, in this case).

 

[epix]

Homeomorphism is a bidirectional function which enables deformation of a given space without making separations of that space.

Since separation is essential for the existence of more than one 0-dim element, there can't be a bidirectional function which (enables deformation between 0-dim and 1-dim) AND (also enables the existence of more then one 0-dim element).

How about providing a rigorous proof of your statement? Anything based on your own tailor-made definition of 0-dim object will not be accepted.

 

[doronshadmi]

How about providing a rigorous proof of your statement? Anything based on your own tailor-made definition of 0-dim object will not be accepted.

Proof?

No proof is needed here because separation is an axiom (a self evident truth) that without it multiple 0-dim elements simply do not exist.

Actually Traditional Math uses such axioms ( http://en.wikipedia.org/wiki/Separation_axiom ):

These conditions are given in order of increasing strength: Any two topologically distinguishable points must be distinct, and any two separated points must be topologically distinguishable. Furthermore, any two separated sets must be disjoint, any two sets separated by neighbourhoods must be separated, and so on.

If points x and y of topological space X have the same set of neighbourhoods, then x and y are called topologically indistinguishable simply because x and y is actually the same point (see http://en.wikipedia.org/wiki/Topological_distinguishability ).

In other words, you do not understand even Traditional Math about this subject.

 

[epix]

Proof?

No proof is needed here because separation is an axiom (a self evident truth) that without it multiple 0-dim elements simply do not exist.

Actually Traditional Math uses such axioms ( http://en.wikipedia.org/wiki/Separation_axiom ):

... Since separation is essential for the existence of more than one 0-dim element, there can't be a bidirectional function which (enables deformation between 0-dim and 1-dim) AND (also enables the existence of more then one 0-dim element).

If points x and y of topological space X have the same set of neighbourhoods, then x and y are called topologically indistinguishable simply because x and y is actually the same point (see http://en.wikipedia.org/wiki/Topological_distinguishability ).

In other words, you do not understand even Traditional Math about this subject.

Why do you steer things away from the metric space to the topological space? Can't you read?

... But any separable zero dimensional metric space is homeomorphic to set of reals (in fact, a subset of the Cantor set) and we are done.

Or is it your tutor who missed?

0-dim objects cannot be located without the presence of a-dimensional objects (a>0) on the same plane. And that's where you start...

I proved what you asked me to prove and that's it.

 

[doronshadmi]

0-dim objects cannot be located without the presence of a-dimensional objects (a>0) on the same plane. And that's where you start...

Exactly, there must be co-existence of more than one dimensional space, and as a result of this co-existence the smaller dimensional is local w.r.t the greater dimensional space, where the greater dimensional space is non-local w.r.t the smaller dimensional space.

Since 0-dimensional space is the smallest dimensional space, it is local w.r.t the rest dimensional spaces.

I used Topological Space because your quoted part of Miller's work (which you used, without giving him the needed reference and credit) uses homeomorphism.

Please look at this ( http://en.wikipedia.org/wiki/Topological_space ):

Two spaces are called homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.

Please also look at http://en.wikipedia.org/wiki/Homeomorphism , you will find there "Topological Space" as well.

I proved what you asked me to prove and that's it.

You proved nothing. All you did is to quote a part of Miller's work, without giving him any credit, and without any understanding of what you quoted ( you simply do not understand http://www.internationalskeptics.com/forums/showpost.php?p=7202595&postcount=15472 ).

But any separable zero dimensional metric space is homeomorphic to set of reals (in fact, a subset of the Cantor set) and we are done.

This is the relevant part taken from Miller's work, and it is wrong because:

Two spaces X and Y are homeomorphic if it is possible to transform X into Y by rescaling it, bending it, twisting it but without introducing puncture or a tear

( http://www.ualberta.ca/MATH/gauss/fcm/Manifolds/PncrCnjctr/hmmrphsm.htm )

Tear and Separation is the same concept.

So " ... any separable zero dimensional metric space is" anything but "homeomorphic to set of reals", simply because homeomorphic spaces can't have tears.

Actually " ... any separable zero dimensional metric space is" exactly one and only one point.

 

[doronshadmi]

In order to understand better why separable zero dimensional metric space is not homeomorphic to set of reals, think about this version of Hilbert's Hotel ( http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel ):

Hilbert's Hotel has only one room, so no matter what number of guests wish to stay in it, only one guest actually can do that.

This version is the precise description of the following part of Millers work:

...any separable zero dimensional metric space...

which is defiantly not homeomorphic to set of reals.

 

[doronshadmi]

Here is another version of Hilbert's Hotel:

Hilbert's hotel has a one room with two entrances, where the room is a 1-dim space and the two entrances are 0-dimensional spaces each.

All the guests that stay in that room are 0-dimensional spaces, such that given any arbitrary pair of guests staying in that room, there is always a room for infinitely many guests between them.

In other words, no matter how many guests are in the room there is always plenty of 1-dim space for more guests.

Conclusion, no amount of guests completely fills Hilbert's hotel.

This version of Hilbert's hotel is the precise description of any arbitrary part of the real-line, which is not less than co-existence of infinity many 0-dim spaces that can't fully fill (or completely cover) Hilbert's hotel 1-dim space room.

 

[The Man]

0-dim is the minimal expression of Locality, such that any other dim > 0 is non-local w.r.t 0-dim, where 0-dim is local w.r.t all dims > 0.

Again if you what a location to only be a point that is up to you.

By generalization, given x-dim < y-dim, x-dim is local w.r.t y-dim and y-dim is non-local w.r.t x-dim., even if x-dim is different than 0-dim.

By your own specifications your “local” is only one point and your “non-local” is more than one point.

You can't get this generalization, exactly as you are unable to understand that not-X is "anything but X", where some particular case of "anything but X" is "the negation of X".

You can’t get that simply typing the word “generalization” doesn’t change what you have specifically claimed.

“not necessarily” is "anything but necessarily" where some case of "anything but necessarily" is "the negation of necessarily".

Looks like you are able to get “not necessarily” only in terms of "the negation of necessarily", but that is your forced limitation of the comprehensive meaning of “not necessarily”, which is "anything but necessarily".

So your “non-local” is “anything but” your “local”? Like your “Emptiness”, “Fullness”, “complex” or "unity"? Each of them is “anything but” your “local”. This also means that your claim of your “non-local” being at least two location is false as “anything but” your “local” doesn’t have to involve any of your “locations”.

You get only the particular case of "the negation of X" of "anything but X", where "anything but X" is the comprehensive meaning of "not-X".

Again, this is your forced limitation and entirely your problem, which is derived directly from your ignorence of the comprehensive meaning of not-X, which is "anything but X.

No Doron it is just the nonsense meaning you want to ascribe that as usual simple contradicts your own claims.

I agree with you. Therefore 0-dim is different than "a lack of dimensions".

So how many dimensions does your point have now?

The Man, you clearly do not distinguish between 0-dim and nothing.

I have made no such assertion and you have been advised of this, so again stop lying.

Wrong. you simply can't comprehend the "anything but ordered" state of thinks that exist at the same level of existence.

So you haven’t got that definition of your “successor” without ordering yet (You could have just said so). Please let us know when you do have it.

Please define the order at the level of nothing, which is "that has no predecessor".

What? Please define your own nonsense assertions if that is what you want?

Another example of your weak reasoning.

Another example of you deliberately ignoring your own notation of ““non-existence”

In this case you are unable to deal with the actuality of nothing as expressed by “ < 0 < x ≤ ∞ < ∞", by forcing “non-existence” expression. In other words you have a fundamental abstraction problem.

In this case you are unable to deal with the actuality of your own “non-existence” notation as expressed by ““non-existence” < 0 < x ≤ ∞ < ∞", by forcing your deliberate ignorance of your “non-existence” expression onto just yourself. In other words you have a fundamental problem with just, well, yourself.

Wrong, you are unable to get Unity, which is beyond any expression (where “successor” is some expression).

So there is “nothing” “beyond” your “fullness” and it has a “successor” that you call “non-existence”.

You still unable the do the needed abstraction in order to get Unity beyond any expression, what so ever.



You still unable the do the needed abstraction in order to get Unity beyond any expression, what so ever.

You’re still unable, apparently deliberately, to get the implications of your own ordering and notations.

Wrong The Man, your weak reasoning simply reflects its own ignorance on the considered subject.

As usual Doron the ignorance remains evidently and apparently deliberately entirely yours.

Now you demonstrate your inability you distinguish between (1) and (1,1).

Nope, evidently that is again just you as I have made no such assertions.

At (1) there is nothing between 1 and 1.

“1 and 1”? You’ve only got “(1)”. Yep it is just you that can’t “distinguish between (1) and (1,1)”

At (1,1) there is redundancy (which is a kind of interval) between 1 and 1.

“(1,1)” is an interval with no difference and nothing between the limits. Which again simply makes your equivocation of “difference” with “interval” and “nothing between” with “difference” demonstrably false.

 

[doronshadmi]

By your own specifications your “local” is only one point and your “non-local” is more than one point.

You can’t get that simply typing the word “generalization” doesn’t change what you have specifically claimed.

Your ignorance of generalization does not change the following:

1) Any smaller dimensional space is local w.r.t to any greater dimensional space.

2) Any greater dimensional space is non-local w.r.t to any smaller dimensional space.

3) 0-dimensional space is the smallest dimensional space and therefore it is local w.r.t the rest of dimensional spaces.

So your “non-local” is “anything but” your “local”? Like your “Emptiness”, “Fullness”, “complex” or "unity"? Each of them is “anything but” your “local”. This also means that your claim of your “non-local” being at least two location is false as “anything but” your “local” doesn’t have to involve any of your “locations”.

You still can't comprehend the involvement of things that save their ids under co-existence, for example:

Stretched 0-dim is not 0-dim anymore, because we get at least 1-dim.

Totally reducible 1-dim is not 1-dim anymore, because we get at most 0-dim.

In other words, there is no homeomorphism among different dimensional spaces.

So how many dimensions does your point have now?

A point is something that has 0 dimension.

I have made no such assertion and you have been advised of this, so again stop lying.

Do you agree that a point is something that has 0 dimension?

So you haven’t got that definition of your “successor” without ordering yet (You could have just said so). Please let us know when you do have it.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=7205990&postcount=15478 .

In this case you are unable to deal with the actuality of your own “non-existence” notation as expressed by ““non-existence” < 0 < x ≤ ∞ < ∞", by forcing your deliberate ignorance of your “non-existence” expression onto just yourself. In other words you have a fundamental problem with just, well, yourself.

The actuality of nothing is no symbol at all, exactly as the actuality of silence is no sound at all.

You can add "actuality" to the words that you don't understand.

So there is “nothing” “beyond” your “fullness” and it has a “successor” that you call “non-existence”.

Now you demonstrate your inability to understand the actuality of "beyond expressions"

Nope, evidently that is again just you as I have made no such assertions.

“(1,1)” is an interval with no difference and nothing between the limits. Which again simply makes your equivocation of “difference” with “interval” and “nothing between” with “difference” demonstrably false.

In other words, you do not understand that nothing between 1 and 1 is resulted by (1), where redundancy between 1 and 1 (which is a kind of an interval) is resulted by (1,1).